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This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. Actually I know such a proof (from Skopets - Zharov book), but it uses the third dimension which ...
Absolute geometry is a geometry based on an axiom system for Euclidean geometry with the parallel postulate removed and none of its alternatives used in place of it. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate.
== Properties ==
It might be imagined that absolute geometry is a rather weak system, but that is not the case. Indeed, in Euclid's Elements, the first 28 Propositions and Proposition 31 avoid using the parallel postulate, and therefore are valid in absolute geometry. One can also...
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Let $X,Y$ are two projective varieties and $f:X\to Y$ is an Iitaka
fibration. Consider the following singular hermitian metric
$$h(\sigma,\sigma)=\left(\int_{X_y}|\sigma|^{\frac{2}{m!}}\right)^{m!}$$
where $y\in Y$ and $\sigma$ is a section of $$\frac{1}{m!}f_*\mathcal
O_X(m!K_{X/Y})|_{...
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